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0^0

askamathematician.com

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Re: 0^0

#41

Post by Bahamut »

na85 wrote: post #35
quchen wrote: post #7

That's a rather long text to say "it's an arbitrary -- and conveniently chosen -- definition of a special case of the power function similar to how 1 is not prime". I also think the presentation was chosen poorly: lots of wrong information before the correct approach is presented. Wikipedia is probably a better source here: https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zer...

My experience has usually been that Wikipedia is an abysmally bad tool for learning mathematics. The articles seem to be written by someone who has zero clue how to teach the concepts and merely is trying to wow the reader with their proof-writing skills.

That is because the majority of Wikipedia's math articles are actually from Planet Math, a website full of math knowledge similar to MathWorld that is geared towards mathematicians.

Re: 0^0

#42

Post by dllthomas »

jomtung wrote: post #39
dllthomas wrote: post #28

Earlier quoted context omitted.

What is "^ 0" in general, that it "makes" 1 "out of" 2, 3, 4, etc?

using the definition in the article where X^n = 1 * X * X ... x^2 = 1 X X x^1 = 1*X x^0 = 1

Yes, for sure. My question was rhetorical, trying to drive intuition away from "But I'm starting with a zero, how do I get anything else?"

Re: 0^0

#43

Post by SilasX »

- "0^0. Why? Because mathematicians said so. No really, it’s true."

- [Detailed explanation of the tradeoffs involved in choosing different definitions of exponentiation.]

So, it's not "because mathematicians said so", it's because of a deep review of the tradeoffs of defining how exponentiation generalizes, the kind of thing that mathematicians happen to study more than other identifiable groups.

Re: 0^0

#44

Post by mason55 »

lovamova wrote: post #40
dinkumthinkum wrote: post #27

Earlier quoted context omitted.

I don't think math is arbitrary at all.

It is. We invented the arabic numerals because they were easy to draw and we could written any numbers with them. Just like we invented higher lever computer languages instead of using assembly. See what Fibonacci used to say in his first book Liber Abaci about using arabic numerals. http://en.wikipedia.org/wiki/Liber_Abaci

The symbols we use to represent math are arbitrary but that doesn't mean the rules behind them are. Many concepts in math are fundamental truths.

Re: 0^0

#45

Post by vlasev »

This is exactly why you have things like 0! = 1, 0 choose 0 = 1, 0^0 = 1, the empty sum is 0 and the empty product is 1, and so on. These are DEFINITIONS and they make notation easier. They basically help the flow of mathematics. It's often difficult to watch someone try to explain "intuitively" why some of these things are the way they are and completely miss the point that they are like this because they help make other things easier.

Re: 0^0

#46

Post by marcosdumay »

kazagistar wrote: post #20
Bahamut wrote: post #15

Earlier quoted context omitted.

Definitions in mathematics are used to make the language describing abstract concepts elegant, as explained in the entry. However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.

Sort of like how we use terse symbols for everything because ink and papyrus is expensive.

Not really. We have some very good reasons for using the definitions we use every day. Those reasons are more related to us than our technology, although our tech is getting so good that we already need to think about it. And by the way, people changed almost all the usual definitions at the XX century.

Now, mathematicians do throw the usual definitions away all the time. It's important to know when to reuse other people's coding, or roll your own.

Re: 0^0

#47

Post by impendia »

didgeoridoo wrote: post #11

Students: Let's come up with some crazy proofs based on our individual levels of understanding. Teachers: Let's do it by the book and come up (somehow) with conflicting answers. Mathematicians: Yeah, sorry guys. We made it all up. Pretty much captures most mathematicians I know.

Well of course we made it up. No one handed us any stone tablets with 0^0 on them.

Mathematicians are always making definitions, and working out which of them should be kept and which should be discarded. We keep the definitions that make the most sense, that make our lives the easiest, that make theorems easy to state, that give math a sense of being natural. Indeed, in the early days of algebraic geometry there were big debates over which definitions to adopt.

It is like deciding on a convention when you design a new programming language. In this case, experience has shown that it is pretty much always better to say that 0^0 is 1 and not 0. Among other reasons, there is exactly one map from the empty set to the empty set.

But if you say 0^0 = 0, you don't get math blowing up in some big contradiction. You just get a little more kludge here and there, a few extra special cases of lemmas that have to be spelled out in more detail. Nothing too awful.

Re: 0^0

#48

Post by vlasev »

forrestthewoods wrote: post #31
quchen wrote: post #7

That's a rather long text to say "it's an arbitrary -- and conveniently chosen -- definition of a special case of the power function similar to how 1 is not prime". I also think the presentation was chosen poorly: lots of wrong information before the correct approach is presented. Wikipedia is probably a better source here: https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zer...

I love Wikipedia. It's amazing. It makes the world a better place. I'm a pretty decent programmer. I do video games so I do lots of 3d math. I'd say I'm decent at that as well. I hate Wikipedia for math. Absolutely hate it. Unless you are a mathematician by trade Wikipedia is damn near useless for learning new math concepts. I don't even bother checking it anymore.

I had an idea for a couple of years now of creating a wikipedia-style mathematics textbook that will be crowd-sourced, standardized and cover all of math in a way that's accessible to learn from on your own. It would have a kind of a zoom function where you can expand details on explanations and calculations to a depth that you prefer. Ideally this kind of thing would start off with basic math and get progressively further into mathematics like the roots of a tree.

Re: 0^0

#49

Post by Bahamut »

ryannevius wrote: post #17

I understand the "math"...the numbers...the work on paper. But how does that translate to something useful in the real world? That, after all, is what useful math helps us do...solve problems for the real, tangible world. Saying that 0^0 = 1 is a cool math game; but translate 0 into something in the real world (i.e. nothing, none, etc.)...and trying to make something out of it other than 0 or "indeterminate" starts t…

The strongest value of math is the most consistently efficient approach for solving non-trivial abstract problems, not necessarily the answer to the question solved by the article. This is due to understanding how to probe a problem, experiment with possibilities, and repeat in a logical manner until you strike gold - it is a skill that is applicable in just about every walk of life.

Re: 0^0

#50

Post by Oculus »

A pretty interesting question(the value of 0^0), in a very understandable format for the layman (me).

Very nice blog post!

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